# Properties

 Label 4-2e18-1.1-c1e2-0-10 Degree $4$ Conductor $262144$ Sign $1$ Analytic cond. $16.7145$ Root an. cond. $2.02196$ Motivic weight $1$ Arithmetic yes Rational yes Primitive no Self-dual yes Analytic rank $0$

# Origins of factors

## Dirichlet series

 L(s)  = 1 − 2·5-s + 10·13-s + 16·17-s + 2·25-s + 6·29-s − 14·37-s + 14·49-s + 18·53-s − 22·61-s − 20·65-s − 9·81-s − 32·85-s − 16·97-s + 18·101-s − 26·109-s − 28·113-s − 10·125-s + 127-s + 131-s + 137-s + 139-s − 12·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯
 L(s)  = 1 − 0.894·5-s + 2.77·13-s + 3.88·17-s + 2/5·25-s + 1.11·29-s − 2.30·37-s + 2·49-s + 2.47·53-s − 2.81·61-s − 2.48·65-s − 81-s − 3.47·85-s − 1.62·97-s + 1.79·101-s − 2.49·109-s − 2.63·113-s − 0.894·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 0.996·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 262144 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 Degree: $$4$$ Conductor: $$262144$$    =    $$2^{18}$$ Sign: $1$ Analytic conductor: $$16.7145$$ Root analytic conductor: $$2.02196$$ Motivic weight: $$1$$ Rational: yes Arithmetic: yes Character: induced by $\chi_{512} (1, \cdot )$ Primitive: no Self-dual: yes Analytic rank: $$0$$ Selberg data: $$(4,\ 262144,\ (\ :1/2, 1/2),\ 1)$$

## Particular Values

 $$L(1)$$ $$\approx$$ $$2.044004972$$ $$L(\frac12)$$ $$\approx$$ $$2.044004972$$ $$L(\frac{3}{2})$$ not available $$L(1)$$ not available

## Euler product

$$L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}$$
$p$$\Gal(F_p)$$F_p(T)$
bad2 $$1$$
good3$C_2^2$ $$1 + p^{2} T^{4}$$
5$C_2$ $$( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
7$C_2$ $$( 1 - p T^{2} )^{2}$$
11$C_2^2$ $$1 + p^{2} T^{4}$$
13$C_2$ $$( 1 - 6 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
17$C_2$ $$( 1 - 8 T + p T^{2} )^{2}$$
19$C_2^2$ $$1 + p^{2} T^{4}$$
23$C_2$ $$( 1 - p T^{2} )^{2}$$
29$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} )$$
31$C_2$ $$( 1 + p T^{2} )^{2}$$
37$C_2$ $$( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
41$C_2$ $$( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} )$$
43$C_2^2$ $$1 + p^{2} T^{4}$$
47$C_2$ $$( 1 + p T^{2} )^{2}$$
53$C_2$ $$( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} )$$
59$C_2^2$ $$1 + p^{2} T^{4}$$
61$C_2$ $$( 1 + 10 T + p T^{2} )( 1 + 12 T + p T^{2} )$$
67$C_2^2$ $$1 + p^{2} T^{4}$$
71$C_2$ $$( 1 - p T^{2} )^{2}$$
73$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
79$C_2$ $$( 1 + p T^{2} )^{2}$$
83$C_2^2$ $$1 + p^{2} T^{4}$$
89$C_2$ $$( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} )$$
97$C_2$ $$( 1 + 8 T + p T^{2} )^{2}$$
$$L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}$$